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Properties

Label 7803.2.a.ca
Level $7803$
Weight $2$
Character orbit 7803.a
Self dual yes
Analytic conductor $62.307$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7803,2,Mod(1,7803)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7803, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7803.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7803 = 3^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7803.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.3072686972\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 6 x^{14} - 3 x^{13} + 76 x^{12} - 69 x^{11} - 354 x^{10} + 523 x^{9} + 720 x^{8} - 1437 x^{7} + \cdots - 51 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} - \beta_{4} q^{5} + (\beta_{10} - \beta_{9} + \beta_{3} - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} - \beta_{4} q^{5} + (\beta_{10} - \beta_{9} + \beta_{3} - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + (\beta_{8} - \beta_{5} - \beta_{4}) q^{10} + ( - \beta_{13} - \beta_{9} + \beta_{6} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{14} + \beta_{13} + \beta_{12} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 6 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 6 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} + 12 q^{22} + 3 q^{23} + 6 q^{25} - 24 q^{26} - 9 q^{28} + 6 q^{29} + 42 q^{32} + 33 q^{35} + 36 q^{38} - 15 q^{40} - 3 q^{43} + 18 q^{44} - 12 q^{46} + 24 q^{47} + 18 q^{49} + 42 q^{50} - 12 q^{52} + 48 q^{53} - 3 q^{55} + 15 q^{56} + 12 q^{58} + 18 q^{59} - 63 q^{62} + 30 q^{64} - 24 q^{65} + 12 q^{67} + 51 q^{70} + 21 q^{71} + 18 q^{73} + 48 q^{74} + 60 q^{76} + 54 q^{77} + 39 q^{79} - 6 q^{80} - 39 q^{82} + 51 q^{83} + 48 q^{86} + 48 q^{88} + 36 q^{89} - 9 q^{91} + 3 q^{92} + 36 q^{94} - 39 q^{95} - 3 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 3 x^{13} + 76 x^{12} - 69 x^{11} - 354 x^{10} + 523 x^{9} + 720 x^{8} - 1437 x^{7} + \cdots - 51 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{14} - 100 \nu^{13} + 257 \nu^{12} + 725 \nu^{11} - 3848 \nu^{10} + 673 \nu^{9} + 17092 \nu^{8} + \cdots - 1818 ) / 73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15 \nu^{14} + 45 \nu^{13} + 326 \nu^{12} - 1111 \nu^{11} - 2152 \nu^{10} + 9147 \nu^{9} + \cdots - 3978 ) / 73 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24 \nu^{14} + 72 \nu^{13} + 507 \nu^{12} - 1690 \nu^{11} - 3414 \nu^{10} + 13657 \nu^{9} + \cdots - 4861 ) / 73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 78 \nu^{14} + 453 \nu^{13} + 279 \nu^{12} - 5602 \nu^{11} + 4271 \nu^{10} + 25460 \nu^{9} + \cdots - 4027 ) / 73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40 \nu^{14} - 339 \nu^{13} + 542 \nu^{12} + 2841 \nu^{11} - 9859 \nu^{10} - 2565 \nu^{9} + 44834 \nu^{8} + \cdots - 6255 ) / 73 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 87 \nu^{14} + 480 \nu^{13} + 460 \nu^{12} - 6181 \nu^{11} + 3009 \nu^{10} + 29970 \nu^{9} + \cdots - 5202 ) / 73 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39 \nu^{14} - 117 \nu^{13} - 833 \nu^{12} + 2801 \nu^{11} + 5639 \nu^{10} - 23169 \nu^{9} + \cdots + 10518 ) / 73 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 123 \nu^{14} + 661 \nu^{13} + 746 \nu^{12} - 8643 \nu^{11} + 2998 \nu^{10} + 43119 \nu^{9} + \cdots - 7274 ) / 73 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 173 \nu^{14} + 1103 \nu^{13} - 41 \nu^{12} - 12249 \nu^{11} + 16727 \nu^{10} + 44920 \nu^{9} + \cdots + 1001 ) / 73 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 134 \nu^{14} + 694 \nu^{13} + 1024 \nu^{12} - 9667 \nu^{11} + 1269 \nu^{10} + 52119 \nu^{9} + \cdots - 11622 ) / 73 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 180 \nu^{14} - 978 \nu^{13} - 1065 \nu^{12} + 12894 \nu^{11} - 5128 \nu^{10} - 64577 \nu^{9} + \cdots + 13134 ) / 73 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 2\beta_{7} + \beta_{6} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{14} + \beta_{9} + 3\beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{3} + 9\beta_{2} + 29\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{14} - \beta_{11} + 2 \beta_{9} + 22 \beta_{7} + 9 \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25 \beta_{14} - \beta_{12} - 2 \beta_{11} + 13 \beta_{9} - 2 \beta_{8} + 42 \beta_{7} + 13 \beta_{6} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{14} + \beta_{13} - 2 \beta_{12} - 16 \beta_{11} - \beta_{10} + 33 \beta_{9} - 5 \beta_{8} + \cdots + 158 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 237 \beta_{14} + 3 \beta_{13} - 15 \beta_{12} - 40 \beta_{11} - 4 \beta_{10} + 138 \beta_{9} + \cdots + 188 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 784 \beta_{14} + 18 \beta_{13} - 36 \beta_{12} - 186 \beta_{11} - 22 \beta_{10} + 385 \beta_{9} + \cdots + 899 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2036 \beta_{14} + 54 \beta_{13} - 164 \beta_{12} - 511 \beta_{11} - 76 \beta_{10} + 1348 \beta_{9} + \cdots + 1479 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6241 \beta_{14} + 218 \beta_{13} - 435 \beta_{12} - 1891 \beta_{11} - 291 \beta_{10} + 3886 \beta_{9} + \cdots + 5581 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16697 \beta_{14} + 653 \beta_{13} - 1600 \beta_{12} - 5391 \beta_{11} - 926 \beta_{10} + 12468 \beta_{9} + \cdots + 11244 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 49078 \beta_{14} + 2253 \beta_{13} - 4465 \beta_{12} - 17807 \beta_{11} - 3065 \beta_{10} + 36228 \beta_{9} + \cdots + 36937 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.07842
−1.81173
−1.78651
−1.05588
−1.04868
−0.494496
0.495172
0.641769
0.701318
1.16033
1.61842
2.01902
2.32100
2.54787
2.77082
−2.07842 0 2.31982 −0.550263 0 −0.555154 −0.664716 0 1.14368
1.2 −1.81173 0 1.28235 0.772083 0 −4.30839 1.30018 0 −1.39880
1.3 −1.78651 0 1.19160 1.94391 0 2.09390 1.44420 0 −3.47281
1.4 −1.05588 0 −0.885120 −0.426533 0 3.34701 3.04634 0 0.450367
1.5 −1.04868 0 −0.900266 −3.57800 0 −2.24665 3.04146 0 3.75218
1.6 −0.494496 0 −1.75547 1.77925 0 3.57140 1.85707 0 −0.879830
1.7 0.495172 0 −1.75481 −0.247583 0 1.35701 −1.85927 0 −0.122596
1.8 0.641769 0 −1.58813 2.68329 0 −4.44365 −2.30275 0 1.72205
1.9 0.701318 0 −1.50815 3.89624 0 0.277307 −2.46033 0 2.73250
1.10 1.16033 0 −0.653644 −2.03861 0 −0.829169 −3.07909 0 −2.36545
1.11 1.61842 0 0.619268 −2.92099 0 0.0208709 −2.23460 0 −4.72737
1.12 2.01902 0 2.07645 −0.707464 0 −4.62694 0.154349 0 −1.42838
1.13 2.32100 0 3.38702 3.93643 0 4.64937 3.21926 0 9.13644
1.14 2.54787 0 4.49165 1.21129 0 0.847998 6.34841 0 3.08621
1.15 2.77082 0 5.67743 −2.75304 0 −2.15490 10.1895 0 −7.62818
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7803.2.a.ca yes 15
3.b odd 2 1 7803.2.a.bx ✓ 15
17.b even 2 1 7803.2.a.bz yes 15
51.c odd 2 1 7803.2.a.by yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7803.2.a.bx ✓ 15 3.b odd 2 1
7803.2.a.by yes 15 51.c odd 2 1
7803.2.a.bz yes 15 17.b even 2 1
7803.2.a.ca yes 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7803))\):

\( T_{2}^{15} - 6 T_{2}^{14} - 3 T_{2}^{13} + 76 T_{2}^{12} - 69 T_{2}^{11} - 354 T_{2}^{10} + 523 T_{2}^{9} + \cdots - 51 \) Copy content Toggle raw display
\( T_{5}^{15} - 3 T_{5}^{14} - 36 T_{5}^{13} + 99 T_{5}^{12} + 480 T_{5}^{11} - 1194 T_{5}^{10} + \cdots - 321 \) Copy content Toggle raw display
\( T_{7}^{15} + 3 T_{7}^{14} - 57 T_{7}^{13} - 149 T_{7}^{12} + 1206 T_{7}^{11} + 2562 T_{7}^{10} + \cdots + 153 \) Copy content Toggle raw display
\( T_{11}^{15} + 6 T_{11}^{14} - 60 T_{11}^{13} - 277 T_{11}^{12} + 1710 T_{11}^{11} + 4194 T_{11}^{10} + \cdots - 459 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 6 T^{14} + \cdots - 51 \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( T^{15} - 3 T^{14} + \cdots - 321 \) Copy content Toggle raw display
$7$ \( T^{15} + 3 T^{14} + \cdots + 153 \) Copy content Toggle raw display
$11$ \( T^{15} + 6 T^{14} + \cdots - 459 \) Copy content Toggle raw display
$13$ \( T^{15} + 3 T^{14} + \cdots - 31393 \) Copy content Toggle raw display
$17$ \( T^{15} \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 152716257 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 1344396147 \) Copy content Toggle raw display
$29$ \( T^{15} - 6 T^{14} + \cdots - 6963651 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 226446443 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 19065212479 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 214469576469 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 4218833799 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 147921471 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 168338485467 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 23134372497 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 13098034421 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 113347799 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 209016345999 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 52394497641 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 69508345077 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 250178061537 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 3817281237669 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 40003405721 \) Copy content Toggle raw display
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